# Japan

### Mathematics From Japan

Not much is known about the early development of mathematics in Japan. There are known connections to Greece and China in that the Japanese adopted a system of exponentional notation like that of Archimedes and imported Chinese texts on arithmetic, algebra, and geometry. Between 1639-1854, Japan self-imposed an isolated existence from the world. Although shut out from the rest of the world, the mathematics community left open their minds and their hearts. Samurai, merchants and farmers would solve geometry problems of the kind that the Greek “father of geometry,” Euclid, became most famous for. After solving these problems, the samurai, merchants and farmers would inscribe their work on tablets and hang them from the roof of temples for all religious worshipers to admire and challenge themselves with. The topics that they covered included geometry, volumes (calculus), and Diophantine problems (visit Greece to learn more). Accordingly, this collection of problems on these “mathematical tablets” became known as sangaku or Japanese Temple Geometry.

### Problem 1

In this problem, from an 1803 sangaku found in Gumma Prefecture, the base of an isosceles triangle sits on a diameter of the large green circle. This diameter also bisects the red circle, which is inscribed so that it just touches the inside of the green circle and one vertex of the triangle, as shown. The blue circle is inscribed so that it touches the outsides of both the red circle and the triangle, as well as the inside of the green circle. A line segment connects the center of the blue circle and the intersection point between the red circle and the triangle. Show that this line segment is perpendicular to the drawn diameter of the green circle.

### Problem 2

This beautiful problem, which requires no more than high school geometry to solve, is written on a tablet dated 1913 in Miyagi Prefecture. Three orange squares are drawn as shown below in the large, green right triangle. How are the radii of the three blue circles related?

### Problem 3

Here is a simple problem (see image below) that has survived on an 1824 tablet in Gumma Prefecture. The orange and blue circles touch each other at one point and are tangent to the same line. The small red circle touches both of the larger circles and is also tangent to the same line. How are the radii of the three circles related? Let r1 be the radius of the small circle, r2 be the radius of the medium circle and r3 be the radius of the large circle. Don't use spaces in your answer. Also your answer will involve square roots so here's an example that demonstrates how you can express the square root of r1:

sqrt(r1)

[Hint: Visit Greece to learn about a famous theorem that has it's origins in China and Babylonia]

### Math Haikus

In addition to the development of mathematics during its period of isolation, a specific type of poem from Japan with three lines, the haiku , also emerged. The first and third lines have 5 syllables with the middle line having 7 syllables.

Pi - ratio of
Around: across a circle -
An endless number?

e^(pi*i)
the intertwining of
two worlds is shown here

### Interactive Japanese Temple Geometry (Sangaku)

www.cyberuk.net/sangaku

### The Earth with a Belt

Imagine the earth was a perfect sphere. Now, imagine there is a giant belt looped around the earth, which fits snugly. Now just like after a big meal, you let the belt out a bit so that it is a little looser. In fact, imagine you loosened it by exactly 2 metres and then pulled it away from the earth evenly. The circumference of the earth is so huge compared to a lousy 2 metres – would there be enough slack, do you think, for a mouse to walk under the belt? How much slack is there?

Try solving the International Color Challenge!

References:

Rothman, T. (1998, May). Japanese Temple Geometry [Online]. Available:
http://www2.gol.com/users/coynerhm/0598rothman.html
http://www.e-poems.org/pi.html
http://www.livejournal.com/~vibrato/38640.html